Solve for $x$ and $y$ using elimination. $\begin{align*}-4x+4y &= -4 \\ 5x+4y &= -8\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}4x-4y &= 4\\ 5x+4y &= -8\end{align*}$ Add the top and bottom equations. $9x = -4$ Divide both sides by $9$ and reduce as necessary. $x = -\dfrac{4}{9}$ Substitute $-\dfrac{4}{9}$ for $x$ in the top equation. $-4( -\dfrac{4}{9})+4y = -4$ $\dfrac{16}{9}+4y = -4$ $4y = -\dfrac{52}{9}$ $y = -\dfrac{13}{9}$ The solution is $\enspace x = -\dfrac{4}{9}, \enspace y = -\dfrac{13}{9}$.